Introduction

The purpose of the book is to deliver a synergetic description of the theoretical background in layered media Green’s functions, integral equation formulations with such Green’s functions, and numerical methods used for their evaluation. To reach this goal, the book comprises three parts. First nine chapters are dedicated to the foundations of electromagnetic theory, complex analysis of Sommerfeld integrals, derivation of integral equations for objects in layered media, and their mixed-potential formulations. Subsequent Chapters 10–13 are dedicated to the method of moments (MoM) discretization of the integral equations. The third part, dedicated to the numerical techniques for evaluation of the layered media Green’s functions and their use in the MoM, is covered in Chapters 14–18. Extensive appendices accompany the chapters and provide background as well as complementary information, which is necessary for understanding of the material, but may already be familiar to the readers depending on their background.

Chapter 1 of the book covers foundational equations of electromagnetism, such as Maxwell equations in time and spectral domains, the second-order partial differential equations (PDEs) governing the electric and magnetic fields, such as curl–curl equations and Helmholtz equations, as well as the boundary conditions. It is also meant to facilitate understanding of the static solutions commonly used in capacitance and inductance extractors. Hence, decomposition of the electromagnetic field at low frequencies into electrostatic and magnetostatic components is outlined. A brief discussion of the key circuit equations, such as Kirchhoff’s current and voltage laws as well as constitutive relations on basic lumped circuit components as a limiting case of Maxwell equations at low frequencies, is covered in this chapter also. The concept of vector and scalar potentials is heavily used throughout this book. These potentials and Helmholtz equations governing them are presented in Chapter 1.

Representations of scalar and dyadic Green’s functions for the fields and potentials in free space are introduced in Chapter 2. Spectral formulations of such free space Green’s functions serve as a basis for the construction of their layered media counterparts handled in the subsequent chapters. The chapter starts from the derivation of 1D Green’s function in the space domain as the description of the electromagnetic field produced by an infinite sheet of unidirectional time-harmonic current. The expression is obtained directly as a solution of the second order ordinary differential equation (ODE) with -function in the right-hand side and also by casting the solution of the ODE into the form of an algebraic equation with respect to its spectrum. The spectral domain solution approach is subsequently repeated for obtaining expansion of the 3D scalar Green’s function over the inhomogeneous plane waves and its decomposition over cylindrical and conical waves known as Sommerfeld identity. The scalar 3D Green’s function is subsequently related to the dyadic Green’s function of the vector potential through the identity dyadic. Construction of Green’s functions of the electric and magnetic field as a solution of the dyadic curl–curl equations concludes this chapter.

Chapter 3 uses Green’s theorems, second-order PDEs for the fields, Green’s functions, and the boundary conditions to formulate the equivalence principle for the field representation in multilayered media. The first part of this chapter deals with scalar problems of electrostatic fields in the presence of arbitrary dielectric and composite conductor/dielectric objects embedded in layered media. Derivation of the equivalence principle is performed through the layer-by-layer application of Green’s theorem, bringing it to the form amenable to the formation of the coupled integral equations with respect to the unknown potential on dielectric boundaries and charges on conductor boundaries. Once the derivation patterns for the scalar integral equations with layered media kernels are established in the simpler case of the electrostatic fields, the chapter reviews key reciprocity relations for the full-wave dyadic Green’s functions. It then proceeds to treat the equivalent integral representations for the electromagnetic fields and classical integral equations for the electric, magnetic, and combined fields in layered media. Other traditional superpositions of the surface integral equations forming the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) and Muller formulations are described. Volume equivalence principle and integral equations allowing to handle continuously inhomogeneous objects situated in layered media are derived using the concepts of the dyadic Green’s function and superposition by introducing polarization currents. Overview of the three classes of single-source surface and volume-surface integral equations concludes this chapter.

Chapter 4 treats the vertical and horizontal dipole radiation problems in the presence of layered media. Spectral representations of the vector potential components are invoked to both formulate the 1D BVPs governing their spectra, demonstrate analytic solutions for these components in the spectral domain, and state the Sommerfeld integral representations for them in the space domain. Obtaining the spectra through ray tracing is illustrated to highlight the physical interpretation of the fields. Non-uniqueness in the definition of the vector potential describing the uniquely defined fields of a horizontal dipole in layered media is explained in detail through step-by-step derivations. The concept of quasi-static image extraction from Green’s function as a way to accelerate the convergence of the Sommerfeld integrals is introduced in this chapter. Multi-valued complex functions formalism attributed to the fields spectra in open layered media and methods for dealing with it using concepts of Riemann surfaces and branch cuts finishes the chapter. It provides a bridge to the subsequent chapters on integration techniques.

Chapter 5 illustrates the evaluation of the Sommerfeld integrals through their reduction to the integrals along the branch cuts using the Cauchy theorem. Freedom and constraints in the definitions of the branch cuts and corresponding Riemann surfaces are illustrated using two classical choices. The first choice deals with hyperbolic branch cuts originally introduced by . Sommerfeld. The second choice is that of the straight branch cuts based on an alternative convention in the definition of the two-valued square root functions entering into open media Green’s function spectra. The original Sommerfeld integration path (SIP) following the real axis on the radial wavenumber plane is shown to be the path of preference in the extreme case when observation and source locations feature large separation along direction of stratification and small lateral separation. The integration along the branch cuts is shown to be preferred in the extreme cases of slight vertical and large lateral separations. These scenarios are subsequently revisited in Chapter 6 as the extreme cases of the steepest descent path (SDP), which varies with the change of mutual location of the source and observation points and provides the non-oscillatory fastest attenuating behavior of the integrand. The Sommerfeld problem of vertical electric dipole radiation in the presence of a half-space is revisited in the context of the branch cuts choice freedom. The illusive presence of the Zenneck surface wave in the solution is demystified. It is shown that while the surface wave pole responsible for the presence of the surface wave (Zenneck wave) in the solution may or may not appear on the physical sheet of the Riemann surface under the Sommerfeld choice of hyperbolic branch cuts, it never shows up on the physical sheet of the Riemann surface under the choice of the straight branch cuts. Visualizations of the contributions into the total field in the Sommerfeld problem for two different choices of the branch cuts are provided to facilitate understanding of its composition. The chapter is concluded with a graphical solution illustrating the formation and the structure of the surface and leaky waves in the canonical problem of dipole radiation above a dielectric slab.

Evaluations of the layered media Green’s functions through evaluation of the integrals along the SDP on the complex plane of radial wavenumber are introduced and explained in detail in Chapter 6. The exponentially varying part of Green’s function spectrum is identified and analyzed to obtain the definition of the path on which the spectrum experiences minimal oscillations and the fastest exponential attenuation. It is shown that while the integration along the SDP was traditionally used for obtaining the analytic form for the far fields, it can be also used for definition of the fields in the near and intermediate zones through numerical integration. The complications in integration along the SDP arise, however, due to the fact that, while the branch cuts and the Riemann sheets defining the integrand remain fixed for a given configuration of the layered media and time-harmonic frequency, the SDP changes in shape depending on the mutual position of the observation point with respect to the point source. This forces the SDP to reside on different sheets of the Riemann surface depending on the location of the observation point...